Counting domino trains

Domino Tilings of the Torus

We consider the problem of counting and classifying domino tilings of a quadriculated torus. The counting problem for rectangles was studied by Kasteleyn and we use many of his ideas. Domino tilings of planar regions can be represented by height functions; for a torus given by a lattice L, these functions exhibit arithmetic L-quasiperiodicity. The additive constants determine the flux of the tiling, which can be interpreted as a vector in the dual lattice (2L)*. We give a characterization of the actual flux values, and of how corresponding tilings behave. We also consider domino tilings of the…

Sandpiles and Dominos

We consider the subgroup of the abelian sandpile group of the grid graph consisting of configurations of sand that are symmetric with respect to central vertical and horizontal axes. We show that the size of this group is (i) the number of domino tilings of a corresponding weighted rectangular checkerboard; (ii) a product of special values of Chebyshev polynomials; and (iii) a double-product whose factors are sums of squares of values of trigonometric functions. We provide a new derivation of the formula due to Kasteleyn and to Temperley and Fisher for counting the number of domino tilings of …

阿兹特克钻石的对称多米诺骨牌铺砌

Symmetric Domino Tilings of Aztec Diamonds

该研究提出归纳求和公式来计算阿兹特克钻石不同对称类型的多米诺骨牌铺砌数，同时得到无约束情形的公式，证明基于作者新提出的图完美匹配计数技术。

Hitting Matrix and Domino Tiling with Diagonal Impurities

As a continuation to our previous work [9, 10], we consider the domino tiling problem with impurities. (1) if we have more than two impurities on the boundary, we can compute the number of corresponding perfect matchings by using the hitting matrix method[4]. (2) we have an alternative proof of the main result in [9] and result in (1) above using the formula by Kenyon-Wilson [6, 7] of counting the number of groves on the circular planar graph. (3) we study the behavior of the probability of finding the impurity at a given site when the size of the graph tends to infinity, as well as the scalin…

Generalized domino-shuffling

The problem of counting tilings of a plane region using specified tiles can often be recast as the problem of counting (perfect) matchings of some subgraph of an Aztec diamond graph A_n, or more generally calculating the sum of the weights of all the matchings, where the weight of a matching is equal to the product of the (pre-assigned) weights of the constituent edges (assumed to be non-negative). This article presents efficient algorithms that work in this context to solve three problems: finding the sum of the weights of the matchings of a weighted Aztec diamond graph A_n; computing the pro…

计数巧合的计算复杂度

Computational complexity of counting coincidences

该研究聚焦组合对象计数的巧合判定问题，证明两类多米诺铺砌的巧合问题不在多项式谱系内（除非谱系坍缩），并拓展至多类组合对象的相关研究与猜想。